math

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RELIABILITY THEORY, MAINTABILITY, AVAILABILITY
THE FAILURE DISTRIBUTION

• PRESENTED BY
• Najuka Jagtap (121417015)
• Pooja Patil (121417014)
• Pallavi Patil(121416009)
• Pratiksha Dalwale(121416005)

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Introduction to Reliability Theory

• DEFINATION
• Reliability is defined
  as probability of a system will function over a
  time period t.
• Generally defined as the ability of a product to
  perform as expected over time

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Reliability Function

• Reliability of a system is expressed as,
• R(t)P (T t)
• Where R(t) 0, R(0)1 ,
• and lim t?8R(t)0
• For given time t, R(t) is the probability that
• the time to failure is greater or equal to t.

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Series configuration

• For a series systems, the reliability is the
  product of the individual components

RS P(E1 n E2) R1
. R2
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• Q. Consider a four component system of which the
  components are independent and identically
  distributed with CFR. If Rs0.95 is the specified
  reliability, find the individual component MTTF.
• Ans. ?0.000128
• MTTF7812.5

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Parallel configuration
1
2
RS P(E1 u E2)
1-(1-R1).(1-R2)
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• Q. Two parallel ,identical and independent
  components have CFR. If it is desired that Rs
  (1000)0.95,find the component and system MTTF.
• Ans. ? 0.000253
• MTTF3952
• MTTFS5928.9

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Properties of Reliability

1. 0 R(t) 1
2. R(0) 1 R(8) 0
3. R(t) is a decreasing function of t.

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What is Maintainability?
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Maintainability definitions

• Maintainability is the probability that a failed
  system will be restored to specified performance
  within a stated period of time when maintained
  under specified conditions
• or
• Maintainability is an inherent design
  characteristic of a system or product and it
  pertains to the ease, accuracy, safety, and
  economy in the performance of maintenance
  actions.

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Objective of Maintainability

• Design and develop systems/equipment which can be
  maintained
• in the least time, at the least cost, and with a
  minimum expenditure of support resources, without
  adversely affecting the item performance or
  safety characteristics
• Maintainability greatly influences reliability
  and availability of a system or subsystem.
• Maintainability must be addressed early in the
  design stage to prevent or reduce failure or down
  times of the system.

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Maintenance Categories

• Failure detection
• Failure isolation
• Repair
• Functional test

• Test of all relevant functions,
• Inspect to detect hidden failures
• Service to replace consumables
• Activities to compensate for drift andto reduce
  wear out failures
• Overall to increase useful life
• Time Change
• Prognostics health management monitor and repair
  before failure

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Maintenance downtime

• Maintenance downtime That portion of downtime
  which can be attributed to preventive- and
  corrective-maintenance functions. Maintenance
  downtime may be expressed in a measure of central
  tendency (arithmetic mean, geometric mean,
  median, and mode). It may also be expressed in
  terms of a maximum value relative to a percentile
  point of distribution of downtime. Symbols of
  maintenance downtime are
• Mct Mean active corrective-maintenance time
  (arithmetic mean). Equal to MTTR.
• Mpt Mean active preventive-maintenance time
  (arithmetic mean).

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• MDT Maintenance downtime (total time during
  which an equipment item is not in condition to
  perform its intended function). MDT includes
  logistics time and waiting or administrative
  time.
• Logistics or Supply Time That portion of
  nonactives maintenance time during which
  maintenance is delayed slowly because a needed
  item is not immediately available.
• Wait or Administrative Time That portion of
  nonactive maintenance time that is not included
  in logistics or supply time.

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Down time analysis

• total down time
• Repair time

Supply delay maintenance access
Diagnosis Replacement or Verification
delay delay
Repair alignment
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The Repair time distribution

• T time to repair a failed system
• h(t) proballity density function
• T the repair will be accomplished within time t
• Then a cumulative distribution function is H(t)
  ?h(t). dt (over the limit of 0 to t)
• Now mean time to repair (MTTR)?th(t). dt
  (over the limit 0 to 8)
• And hence variance v(?(t-MTTR)2 . H(t) . dt )

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Example

• The mean time to repair a failed widget has the
  following probability density function
  h(t)0.07662t 1 t 5hr.
• solution
• H(t)?0. 07662t .dt (limit 1 to t)
• 0.03931t2 - 0. 03931
• The probability of completing a repair in less
  than 3hr is
• 0. 0393132-0. 03931
• 0.31448
• MTTR ?t(0. 07662 t2).dt (limit 1 to 5)
• 3.16hr

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• There are also other methods for calculating
  Repair time
• Exponential process
  H(t)1 -
  exp(-t/MTTR)
• 2. Lognormal Repair times
• (1/v(2PIts)exp(-0.5 ln( t/tmed )2)/s2) ,
  t0

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Problem on exponential repair time

• H(t)1 - exp(-t/MTTR)
• Example A component repaired at the constant
  rate of 10per 8 hr day what is the probability
  for single repair exceding1 hr?
• Solution MTTR 0.1 day0.8hr
• therefore Pr(tgt1)1-H(t)exp(-1/0
  .8)0.2865

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What is AVAILABILITY?
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Availability

• Availability- The probability that a system is
  operating under given conditions at a given
  instant of time.
• Availability depends on both reliability and
  maintainability.
• For system availability both failure probability
  distributions must be considered.

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Classification of Availability
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• Instantaneous Availability A(t) is the
  probability that system is operational at any
  arbitrary time t.It it given by expected up time
  of the system.
• A(t)Ezt
  ...1.
• Where z(t) is indicator variable defined as
• Z(t)0 if system is in operating
  state at time
• Z(t)1 if system is in failed state
  at time
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  2.
• In view of eq 1 eq 2 can be rewritten as
• A(t)(1)Pz(t)0 (0)Pz(t)1
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• Average up-time availability A(T) it is defined
  as proportion of time during which system is
  available for use in a specified interval 0,T
• A(T)1/T ? A(t) dt (over the limit of 0
  to T) .4.

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• .

 
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Examples
1. If we are using equipment which has a mean
time between failure (MTBF) of 81.5 years and
mean time to repaire (MTTR) of 1 hour Solution
MTBF in hours 81.536524713940 (This is a
reliability parameter and often has a high level
of uncertainty!) Inherent Availability (Ai)
MTBF/(MTBFMTTR)
713940/713941
99.999859
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2. A system has mean time to failure of 150
hours a mean time to repaire is 20 hours what
is the steady state availability of system ?
Solution-Steady state availability of system
isA(8) MTTF/(MTTFMTTR)150/(15020)0.8824

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The Failure Distribution

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1) The Reliability Function

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The Reliability Function
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2) The Cumulative Distribution Function

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The Cumulative Distribution Function
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3) The Probability Density Function

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Area 1.0
The Probability Density Function
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Relation between R(t), CDF F(t) PDF f(t)

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4) Hazard Rate Function

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Relation between R(t) and ?(t)

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Types of Failure Distribution

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Problems

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Problems

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THANK YOU!!!!